| L(s) = 1 | − 0.765·2-s + 3-s − 0.414·4-s − 1.84·5-s − 0.765·6-s + 1.08·8-s + 9-s + 1.41·10-s + 0.765·11-s − 0.414·12-s − 13-s − 1.84·15-s − 0.414·16-s − 0.765·18-s + 0.765·20-s − 0.585·22-s + 1.08·24-s + 2.41·25-s + 0.765·26-s + 27-s + 1.41·30-s − 0.765·32-s + 0.765·33-s − 0.414·36-s − 39-s − 2·40-s + 1.84·41-s + ⋯ |
| L(s) = 1 | − 0.765·2-s + 3-s − 0.414·4-s − 1.84·5-s − 0.765·6-s + 1.08·8-s + 9-s + 1.41·10-s + 0.765·11-s − 0.414·12-s − 13-s − 1.84·15-s − 0.414·16-s − 0.765·18-s + 0.765·20-s − 0.585·22-s + 1.08·24-s + 2.41·25-s + 0.765·26-s + 27-s + 1.41·30-s − 0.765·32-s + 0.765·33-s − 0.414·36-s − 39-s − 2·40-s + 1.84·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6983256049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6983256049\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.765T + T^{2} \) |
| 5 | \( 1 + 1.84T + T^{2} \) |
| 11 | \( 1 - 0.765T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.84T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 1.84T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 0.765T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.84T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 + 0.765T + T^{2} \) |
| 89 | \( 1 + 0.765T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.112621342824537068419623624455, −8.699980067180465745430238685285, −7.75913918712557634616348723274, −7.58685526756450588147138013593, −6.76791483618631078616987492046, −4.99975660184811136413196184136, −4.15010069030936891766137573910, −3.77461314160796408340147850540, −2.51744758282499395694981685474, −0.920097190925612667886957864397,
0.920097190925612667886957864397, 2.51744758282499395694981685474, 3.77461314160796408340147850540, 4.15010069030936891766137573910, 4.99975660184811136413196184136, 6.76791483618631078616987492046, 7.58685526756450588147138013593, 7.75913918712557634616348723274, 8.699980067180465745430238685285, 9.112621342824537068419623624455
